Tan, Cos, SinDate: 05/21/99 at 17:30:09 From: Siobhan Subject: tan, cos, sin Please tell me about tan, sin, and cos. Also, is there a way of working them out without a calculator? How do you know which one to use? Date: 05/22/99 at 16:19:49 From: Doctor Jeremiah Subject: Re: tan, cos, sin Hi there. Trigonometry is complicated until you understand it. Then you wonder why it seemed so complicated. First you need a triangle, but not just any triangle. You need a triangle that has one angle of 90 degrees. + /| / | / | / | / | / | / | c b / | / | / | / x 90| +------a-----+ I have labelled the sides a, b and c. I also labelled one angle as x. The sine of the angle x is written as sin(x). The cosine of the angle x is written as cos(x). The tangent of the angle x is written as tan(x). Each of these values is just a fraction made of the lengths of two of the triangle's sides. For example: sin(x) is a fraction made by taking the length of the short side opposite x and dividing it by the long side. Written as an equation sin(x) = b/c cos(x) is a fraction made by taking the length of the short side next to x and dividing it by the long side. Written as an equation cos(x) = a/c tan(x) is a fraction made by taking the length of the short side opposite x and dividing it by the length of the short side next to x. Written as an equation tan(x) = b/a Let's say that we know the lengths of two sides of the triangle below: + /| / | / | / | / | / | / | / | 12 / | / | / | / x 90| +------------+ 5 We can find the tangent of the angle x. This we can do because tan(x) is a fraction made by taking the length of the short side opposite x (12) and dividing it by the length of the short side next to x (5). tan(x) = 12/5 = 2.4 Let's say that we know only one side of the triangle but that we know the angle as well: + /| / | / | / | / | / | / | c | 12 / | / | / | / 30 90| +------a-----+ Here we can find the lengths of either side. - If we use the equation for the sine of x, then when we put the angle and the opposite side's length in we get sin(30) = 12/c, and after rearranging we would get c = 12/sin(30). - If we use the equation for tangent, then when we put the angle and the opposite side in we get tan(30) = 12/a, and after rearranging we would get a = 12/tan(30). The problem here is that we don't know the value of the sine of 30 degrees. Can we figure it out without a calculator? Sure. There are several ways, but they are complicated to explain. It turns out that the sine of 30 is 0.5. One method of calculating sines and cosines is to use a Taylor series. The tangent of an angle is calculated differently. The tangent is just the sine divided by the cosine. Let me know if you need more help. - Doctor Jeremiah, The Math Forum http://mathforum.org/dr.math/ |
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